Cut off the stem inside the portobellos. With a spoon, scrape out and discard the dark gills. Place the mushrooms in a shallow pan, tops down. Pour 2 tablespoons of the dressing in each mushroom cavity and turn them over a few times so all sides are coated with the marinade.
Heat the oven to 350° F. Heat a large ovenproof skillet over medium-high flame. Add a tablespoon of oil and swirl the pan so the oil coats the bottom of the pan with a thin film. Slip the mushrooms into the pan, tops down. Sear 5 minutes, or until they take on a rich golden tan. Season with a pinch of salt and freshly ground pepper.
Turn the mushrooms so the seared side is up. Place the skillet in the oven and roast 15 to 20 minutes. Remove from the oven and transfer the mushrooms to a plate. Set aside the skillet, keeping the marinade and roasting juices in the pan.
Slice the bread 1/2 inch thick. You’ll need 4 (~8 x 3-inch) slices for this recipe. Cut the red pepper into 4 (2 ½ inch-wide) portions.
Build the sandwiches. Arrange two slices of bread, each with 2 slices of mozzarella cheese. Top each slice of bread with 2 roasted mushrooms followed by 2 portions of red pepper. Scrunch as much arugula as you can manage over the peppers, and then top with the remaining cheese and bread.
Return the skillet to the stovetop over medium-high flame. Melt a dollop of butter into the oil and vinegar marinade/mushroom juices remaining in the pan.
Place the sandwiches in the skillet, pressing gently on the tops for a minute or so. Cook 3 or 4 minutes, until golden. Carefully flip the sandwich to the other side and cook another 3 or so minutes, until the cheese melts and the bread is crispy golden.
Cut in half and serve immediately.